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On common zeros of Legendre's associated functions

Author: Norbert H. J. Lacroix
Journal: Math. Comp. 43 (1984), 243-245
MSC: Primary 33A45
MathSciNet review: 744933
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Abstract: In this paper it is proved that any two given Legendre associated functions $ P_n^m(\mu )$ and $ P_n^s(\mu )$, where $ n \geqslant 1$ is an integer and where one of the integers m or s may be 0 (and $ m \ne \pm s$), have either no zero in common or exactly one common zero, namely $ \mu = 0$. An auxiliary result states that the $ n - m$ zeros of $ P_n^m$ known to lie in the open interval $ ( - 1,1)$ lie in fact in the open interval $ ( - c,c)$, where $ \pm c$ are the two zeros of $ n(n + 1) - {m^2}/(1 - {\mu ^2})$ which is one of the coefficients in the Legendre associated equation satisfied by $ P_n^m$. Some monotonicity behavior of $ P_n^m$ is simultaneously described.

The proof of the main result is based on properties of Prüfer polar coordinates.

References [Enhancements On Off] (What's this?)

  • [1] B. C. Goodwin & N. H. J. Lacroix, "A further study of the holoblastic cleavage field," J. Theoret. Biol. (To appear.)
  • [2] E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea, New York, 1955. MR 0064922 (16:356i)
  • [3] H. Sagan, Boundary and Eigenvalue Problems in Mathematical Physics, Wiley, New York, 1966. MR 0118932 (22:9701)

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Keywords: Legendre functions, associated Legendre functions, Legendre's associated functions, zeros of orthogonal functions
Article copyright: © Copyright 1984 American Mathematical Society

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